Distributed coordinated attitude tracking control of a multi-spacecraft system with dynamic leader under communication delays

This paper is dedicated to the challenging issue of the cooperative attitude tracking control of a multi-spacecraft system under communication delays. A state estimator using the attitude information of the neighbors is designed for each follower spacecraft to estimate the time-varying attitude information of the leader spacecraft in the case that the leader spacecraft cannot directly communicate with all following spacecraft. By constructing auxiliary variables based on estimated values and proposing a fixed-time control law to ensure that the auxiliary variables of the spacecraft can reach zero in fixed time, which is independent of the initial state, the effects of time-varying reference attitudes on the system can be reduced. The attitude error and the estimation error are proven to converge to the region containing the origin by input-to-state stability theory combined with the Lyapunov–Krasovskii approach. To further illustrate the effectiveness of the proposed control algorithm, numerical simulation results are presented.

Background Euler-Lagrange system. Consider a multiple rigid spacecraft system consisting of n followers, labeled as spacecraft i, i = 1, . . . , n , and one leader labeled as spacecraft 0. The attitude of the ith follower is described by modified Rodriguez parameters (MRPs) as where σ i = [σ i1 , σ i2 , σ i3 ] ∈ R 3 is the MRP of the ith spacecraft denoting the attitude orientation of the body-fixed frame with respect to the inertial frame. ω i ∈ R 3 is the angular velocity of the ith rigid body with respect to the inertial frame expressed in the body frame of the ith rigid body. G(σ i ) is described as The matrix G(σ i ) is provided with following properties The dynamics of the ith spacecraft in the system are described by where J ∈ R 3×3 is the inertia matrix of the ith spacecraft and u i ∈ R 3 is the control torque of the ith spacecraft. The n followers are represented with Euler-Lagrange equations by combining (1) and (4) as det(G(σ i )) = (1 + σ i 2 ) (4) www.nature.com/scientificreports/ w h e r e C i (σ i ,σ i ) = −G −T (σ i )J i G −1 (σ i )Ġ(σ i )G −1 (σ i ) − G −T (σ i )(J i ω i ) × G −1 (σ i ) , τ i =G −T (σ i )u i , a n d M i (σ i ) = G −T (σ i )J i G −1 (σ i ) . According to (3), M i (σ i ) is the symmetric positive-definite inertia matrix. For more details on the Euler-Lagrangian model, please refer to Meng and Ren 18 . Throughout the subsequent analysis, the following fundamental properties of system (5) are given by Zhai and Xia 17 : Property 1 Parameter Boundedness: For any ith spacecraft, there exist positive constants k m , k M and k C , Property 2 Linearity in the dynamic parameters: for any x, y ∈ R 3 , where i is the constant parameter vector associated with the ith spacecraft and Y i (σ i ,σ i , x, y)� i is a known regression matrix.
Graph theory. Graph theory is briefly introduced to represent the topology of the information flow among multi-spacecraft system. A directed weighted graph can be denoted as G = (N, E, A) , where N = (n 0 , n 1 , . . . , n n ) is a finite nonempty set of nodes and E ⊆ N × N is a set of unordered pairs of nodes. An edge (n i , n j ) ∈ E denotes that the jth spacecraft can obtain the information from the ith spacecraft and vice versa. The adjacency matrix A = [a ij ] ∈ R n×n of graph G is defined such that adjacency elements a ij satisfy if (n i , n j ) ∈ E , and a ij = 0 otherwise. The Laplacian matrix L = [l ij ] ∈ R n×n associated with A is defined as l ii = i� =j a ij and l ij = −a ij , where i = j . The directed graph G has a directed spanning tree if and only if there exists at least one node having a directed path to all other nodes. The leader adjacency matrix is defined as B=diag(a 10 , a 20 , . . . , a n0 ) , where a i0 =1 if the ith spacecraft can communicate with the leader spacecraft and a i0 = 0 otherwise. Define H = L + B.

Assumption 1
The communication topology graph Ḡ of the multi-spacecraft system has a directed spanning tree. Mathematic background. Consider the following system: where x ∈ R n and g : R + × R n → R n is a nonlinear function, which can be discontinuous. For system (6), the following definitions and lemmas are given.
Definition 1 (Polyakov 8 ) The equilibrium of system (6) is fixed-time stable if it is finite-time stable, and the settling time T(x 0 ) is uniformly bounded for any initial states, that is, ∃T max > 0 , such that T(x 0 ) ≤ T max , ∀x 0 ∈ R n . Lemma 2.2 (Polyakov 8 ) If there exists a continuous radially unbounded function V : R n → R + ∪ {0} such that (2) for any solution x(t) of (6) satisfies the inequality D * V (x(t)) ≤ −(αV p (x(t)) + βV q (x(t))) k for some α, β, p, q, k > 0 : pk < 1, qk > 1 , then the origin is globally fixed-time stable for system (6) and the following estimate holds: where

Coordinated attitude tracking control with a dynamic leader under communication delays
In this section, we address the distributed coordinated attitude tracking control problem for multi-spacecraft under directed communication graph with a dynamic leader under communication delays. The attitude of the leader spacecraft is generated as 24 where Q ∈ R m×m and N ∈ R 3×m are constant matrices and v ∈ R m is an auxiliary variable. (11) can generate multiple types of σ 0 , such as step signals of arbitrary amplitude, ramp signals of arbitrary slope, and sinusoidal signals of arbitrary amplitudes and initial phases. By the reasonable design of N, a linear combination of v will generate a variety of different signals 25 . We assume m = 3 in this paper. Define the attitude tracking error and its derivative of each follower spacecraft as where i = 1, 2 . . . , n , and the control objective of this paper is to ensure lim

Remark 1 System
Because of the existence of time delay in communication between spacecraft, the distributed estimator is proposed as is the estimate of attitude information v 0 (t) , α is a positive constant and. T is the time-varying communication delay.

Remark 2
In this paper, the communication time delay between the available spacecraft is considered. There is v i (t − T) in (13), however, this item is not caused by the communication time delay, but the states of the ith spacecraft at the time t − T is called during the controller calculation process.

Assumption 2
The time-varying communication delay T satisfies 0 ≤ T ≤ T 0 and 0 ≤Ṫ(t) ≤ d < 1 , where T 0 is a positive constant and there exist positive definite matrices P 1 , P 2 , W and reasonably matrices Z such that  26 , although the LMI forms are similar, the design of this paper can guarantee each block matrix on main diagonal negative definite, thus guaranteeing the correctness of the proof. We first give the boundedness proof for the relevant states (13). Consider the following Lyapunov-Krasovskii functional where v(t) is the column vector of column v i (t) , and the derivative of V 1 is  (14).

Define auxiliary variables as
where β is a positive constant. We obtain by Property 2, Combining (5) and (18), the coordinated attitude tracking control law for the ith follower spacecraft is proposed as where k 1 and k 2 are positive constants, 0 < p < 1 and q > 1. (5), the auxiliary variable s i of each follower spacecraft converges to zero in fixed time by properly designing the parameter coordinated control law (19).

Theorem 1 Considering the multi-spacecraft attitude system
Proof We have shown that the states v are bounded in (14); moreover, v i , σ ri and σ ri are bounded from (13) and (16). By Property 1, Y i (σ i ,σ i ,σ ri ,σ ri )� i is bounded for bounded states σ ri and σ ri . τ , sand are column stack vectors of τ i , s i and i ,respectively. C(σ ,σ ) , M(σ )and Y are block diagonal matrices of C i (σ i ,σ i ) , M i (σ i ) and Y i , respectively. Substituting (17), (18) and (19) into (5) Since M(σ ) is positive definite, we can obtain V 2 < 0 , which means that s i is bounded. According to the Property 1 and Lemma 2.2, the auxiliary variable s i for the ith spacecraft will converge to zero in fixed time with a settling time The proof of Theorem 1 is complete.
. σ (t) and ṽ(t) are the column stack vectors of σ i (t) and ṽ i (t) , respectively. Rewrite (13) and (16) as Theorem 2 Consider the linear system with communication delays described by (27). If there exist positive definite matrices P 3 , P 4 and P 5 such that the following linear matrix inequality (LMI) holds: then system state x converges to zero asymptotically. Under Assumptions 1 and 2, the distributed estimator (13) and the coordinated control law (19), which depends on (16) and (17), can guarantee that the attitude tracking error of the follower spacecraft asymptotically converges to zero. (28) is inspired by (Polyak et al. 27 ). Similar to the analysis in Remark 3, the corresponding solutions P 3 , P 4 and P 5 can be obtained by the mature solution method of linear matrix inequality (Last 28 ). In LMI (28), the upper bound T 0 of the communication delay that the system can cope with only related to design parameters and not to the structure of the controller itself. Therefore, T 0 does not affect the proof of system stability. But in practice, the size of T 0 affects the selection of α and β.

Remark 5 The form of LMI
Proof Consider the following Lyapunov-Krasovskii functional for system (27): Taking the time derivative of the Lyapunov-Krasovskii function V 3 , we have By employing the Leibniz-Newton formula, we obtain . It can be obtained that the following inequality holds according to Lemma 2.5 Under Assumption 2, the following inequality can be obtained by using (31): By substituting (28) into (32), linear system (27) is asymptotically stable, which means that lim t→∞σ (t) = 0 and lim t→∞ṽ (t) = 0 . Since the linear system (27) is asymptotically convergent, its characteristic roots all have negative real parts, so it is exponentially convergent. It thus follows that when s = 0 , the system given by (24) and (25) is www.nature.com/scientificreports/ globally exponentially stable at the origin σ T ,ṽ T T = 0 . Combining that s(t) has been proven to be bounded and converge to zero in fixed time in Theorem 1, the system given by (24) and (25) is input-to-state stable with respect to the input s(t) from Lemma 2.3. From Theorem 1, V 2 → 0 as t → T s ,i.e., s(t) → 0 as t → T s . Since systems (24) and (25) are input-to-state stable with respect to the input s(t) and the state σ (t) T ,ṽ(t) T T , V 3 → 0 as t → ∞ , i.e., σ (t) → 0 and ṽ(t) → 0 as t → ∞ . This completes the proof of Theorem 2.

Simulation results
In this section, to demonstrate the validity of the proposed control algorithms, numerical simulations for a multi-spacecraft system are conducted, and the communication topology is described by Fig. 1. The system consisting of four follower spacecraft and one leader spacecraft satisfies Assumption 1. Table 1 Figure 5 shows the control torques of the follower spacecraft. The attitude tracking errors σ i (t) , estimation errors ṽ i (t) and auxiliary variable s i (t) can converge to region |σ iI | < 7 × 10 −4 , |s iI | < 6 × 10 −5 in 8 s and |ṽ iI | < 6 × 10 −4 in 5 s. As the desired attitude is dynamic, the control torques will never be zero, as shown in Fig. 5. After the multi-spacecraft system reaches attitude synchronization, the control torque tends to be stable. Although the fixed time control algorithm cannot consistently converge the error of the auxiliary variable to zero, as shown in Fig. 4, it can still be used to reduce the convergence time in the case of tracking the dynamic target, as the convergence time is influenced by the initial conditions in the traditional control process, which continuously change due to the dynamic reference attitude.

Conclusions
This paper studies the problem of distributed coordinated control of spacecraft attitude tracking under timevarying communication delays, where the communication topology is described as a directed graph and the reference attitude is dynamic. Since the reference attitude information is available to only some of the following spacecraft, a distributed estimator that uses neighbor information under communication delay is proposed to estimate the reference attitude information. An auxiliary variable is designed based on the estimated information. By means of the fixed-time control method, a fixed-time control algorithm is proposed that achieves convergence of the auxiliary variable with fixed time, and the convergence time is independent of the initial state. This means that the influence of the constantly changing initial conditions on the system is reduced. By using the Lyapunov-Krasovskii functional approach and input-to-state stability theory, the proposed control algorithms guarantee that the attitude of each following spacecraft can be asymptotically synchronized with the dynamic attitude of the leader spacecraft.